The concept of a closed map should not be confused with that of a closed operator. Besides, an open map need not be closed and vice versa.[2]

Neither open nor closed maps are required to be continuous. Although their definitions seem more natural, open and closed maps are much less important than continuous maps. Recall that, by definition, a function f : X → Y is continuous if the preimage of every open set of Y is open in X. (Equivalently, if the preimage of every closed set of Y is closed in X).

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Every homeomorphism is open, closed, and continuous. In fact, a bijective continuous map is a homeomorphism if and only if it is open, or equivalently, if and only if it is closed.

If Y has the discrete topology (i.e. all subsets are open and closed) then every function f : X → Y is both open and closed (but not necessarily continuous). For example, the floor function from R to Z is open and closed, but not continuous. This example shows that the image of a connected space under an open or closed map need not be connected.

Whenever we have a product of topological spaces X=ΠXi, the natural projections pi : X → Xi are open (as well as continuous). Since the projections of fiber bundles and covering maps are locally natural projections of products, these are also open maps. Projections need not be closed however. Consider for instance the projection p1 : R2 → R on the first component; A = {(x,1/x) : x≠0} is closed in R2, but p1(A) = R − {0} is not closed. However, for compact Y, the projection X × Y → X is closed. This is essentially the tube lemma.

To every point on the unit circle we can associate the angle of the positive x-axis with the ray connecting the point with the origin. This function from the unit circle to the half-open interval [0,2π) is bijective, open, and closed, but not continuous. It shows that the image of a compact space under an open or closed map need not be compact. Also note that if we consider this as a function from the unit circle to the real numbers, then it is neither open nor closed. Specifying the codomain is essential.

The function f : R → R with f(x) = x2 is continuous and closed, but not open.

It is useful to have conditions for determining when a map is open or closed. The following are some results along these lines.

The closed map lemma states that every continuous function f : X → Y from a compact spaceX to a Hausdorff spaceY is closed and proper (i.e. preimages of compact sets are compact). A variant of this result states that if a continuous function between locally compact Hausdorff spaces is proper, then it is also closed.